The Feedback Hamiltonian is the Score Function: A Diffusion-Model Framework for Quantum Trajectory Reversal

TL;DR

This paper reveals that the feedback Hamiltonian in quantum systems acts as the score function of the trajectory distribution, linking quantum control to diffusion models and enabling ML-based trajectory reversal methods.

Contribution

It provides a rigorous proof that the feedback Hamiltonian is the score function, extending the diffusion model analogy to quantum trajectories and multi-qubit systems.

Findings

The feedback Hamiltonian equals the score function of quantum trajectories.

A continuous family of path measures is generated by the feedback gain.

ML score estimation methods can replace analytic formulas in experiments.

Abstract

In continuously monitored quantum systems, the feedback protocol of Garc\'ia-Pintos, Liu, and Gorshkov reshapes the arrow of time: a Hamiltonian $H_{\mathrm{meas}} = r A / \tau$ applied with gain $X$ tilts the distribution of measurement trajectories, with $X < -2$ producing statistically time-reversed outcomes. Why this specific Hamiltonian achieves reversal, and how the mechanism relates to score-based diffusion models in machine learning, has remained unexplained. We compute the functional derivative of the log path probability of the quantum trajectory distribution directly in density-matrix space. Combining Girsanov's theorem applied to the measurement record, Fr\'echet differentiation on the Banach space of trace-class operators, and K\"ahler geometry on the pure-state projective manifold, we prove that $\delta \log P_F / \delta \rho = r A / \tau = H_{\mathrm{meas}}$. The Garc\'ia-Pintos feedback Hamiltonian is the score function of the quantum trajectory distribution -- exactly the object Anderson's reverse-time diffusion theorem requires for trajectory reversal. The identification extends to multi-qubit systems with independent measurement channels, where the score is a sum of local operators. Two consequences follow. First, the feedback gain $X$ generates a continuous one-parameter family of path measures (for feedback-active Hamiltonians with $[H, A] \neq 0$), with $X = -2$ recovering the backward process in leading-order linearization -- a structure absent from classical diffusion, where reversal is binary. Second, the score identification enables machine learning (ML) score estimation methods -- denoising score matching, sliced score matching -- to replace the analytic formula when its idealizations (unit efficiency, zero delay, Gaussian noise) fail in real experiments.